5-4 Direct Variation

Warm Up

1. Regina walked 9 miles in 3 hours. How many miles did she walk per hour?

2. To make 3 bowls of trail mix, Sandra needs 15 ounces of nuts. How many ounces of nuts does she need for 1 bowl of trail mix?

3 mi per hour

5 oz

5-4 Direct Variation

Problem of the Day

Paul has earned $60 from his paper route. Each day he earns $3.50 more. How many days will it take for Paul's earnings to top $100?

12 days

5-4 Direct Variation

Learn to identify, write, and graph an equation of direct variation.

5-4 Direct Variation

Vocabulary

direct variationconstant of variation

5-4 Direct Variation

Direct variation is a linear relationship between two variable that can be written in the form y = kx or k

= , where k 0.

*The fixed number k in a direct variation equation is the constant of variation.

*A line represents a direct variation when it goes through the origin!

yx

5-4 Direct Variation

You can read direct variation as “y varies directly as x” or “y is directly proportional to x” or “y varies with x.”

Reading Math

5-4 Direct Variation

Tell whether each equation represents a direct variation. If so, identify the constant of variation.

y + 8 = x

Additional Example 1A: Identifying a Direct Variation from an Equation

Solve the equation for y. Subtract 8 from both sides.y + 8 = x

– 8 = – 8

y = x – 8

The equation is not in the form y = kx, so y + 8 = x is not a direct variation.

5-4 Direct Variation

Tell whether each equation represents a direct variation. If so, identify the constant of variation.

3y = 2x

Additional Example 1B: Identifying a Direct Variation from an Equation

Solve the equation for y. Divide both sides by 3.

The equation is in the form y = kx, so the original equation 3y = 2x is a direct variation.

y = x 23

Write as x . 2x3

23

3y = 2x3 3

5-4 Direct Variation

Tell whether each equation represents a direct variation. If so, identify the constant of variation.

y + 3 = 3x

Check It Out: Example 1A

Solve the equation for y. Subtract 3 from both sides.

y + 3 = 3x – 3 – 3

y = 3x – 3

The equation is not in the form y = kx, so y + 3 = 3x is not a direct variation.

5-4 Direct Variation

Tell whether each equation represents a direct variation. If so, identify the constant of variation.

4y = 3x

Check It Out: Example 1B

Solve the equation for y. Divide both sides by 4.

The equation is in the form y = kx, so the original equation 4y = 3x is a direct variation.

y = x 34

Write as x . 3x4

34

4y = 3x4 4

5-4 Direct Variation

Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

Additional Example 2A: Identifying a Direct Variation from a Table

Find for each ordered pair. yx

= yx

2 69

= y x

= 3 99

1 33

= yx

4 129

k is not the same for each ordered pair.

The data does not represent a direct variation.

Price (c) 69 99 129

Weight (oz) 2 3 4

5-4 Direct Variation

In a direct variation where k is positive, when x increases, y also increases; when x decreases, y also decreases.

Helpful Hint

5-4 Direct Variation

y= = 2.54 x

2.54 1

Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

Additional Example 2B: Identifying a Direct Variation from a Table

k = 2.54 for each ordered pair.

The data represent a direct variation where k = 2.54. The equation is y = 2.54x

Inches 1 2 5

Centimeters 2.54 5.08 12.70

Find for each ordered pair. yx

= = 2.54 5.08 2

yx = = 2.54 12.7

5yx

5-4 Direct Variation

Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

Check It Out: Example 2A

k is not the same for each ordered pair.

The data does not represent a direct variation.

Price (c) 5 10 15

Weight (lb) 2 3 4

Find for each ordered pair. yx

= 2 5x

y = 3 10x

y = 4 15x

y

5-4 Direct Variation

Tell whether each set of data represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

Check It Out: Example 2B

k = 3 for each ordered pair.

The data represent a direct variation where k = 3. The equation is y = 3x

Meters 3 4 5

Miles 9 12 15

Find for each ordered pair. yx

= = 3 9 3x

y= = 3 12

4xy

= = 3 15 5x

y

5-4 Direct Variation

The graph is a line through (0, 0). This is a direct variation. The Slope of the line is ½, so k = –½ . The equation is y = -½x.

Additional Example 3: Identifying a Direct Variation from a Graph

x

y

0–2–4 2 4

2

4

–2

–4

Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

5-4 Direct Variation

In a direct variation, the slope, k, represents a constant rate of change.

Helpful Hint

5-4 Direct Variation

The line does not pass through (0, 0). This is not a direct variation.

Check It Out: Example 3

x

y

0–2–4 2 4

2

4

–2

–4

Tell whether each graph represents a direct variation. If so, identify the constant of variation and then write the direct variation equation.

5-4 Direct Variation

Lesson Quiz: Part I

Tell whether each of the following represents

a direct variation. If so, identify the constant

of variation.

1. 12y = 6x

2.

yes; k = ½

no

5-4 Direct Variation

Lesson Quiz: Part II

3. A cheetah runs at a speed of 0.75 miles per minute.

a. Write a direct variation equation for the distance

y the cheetah runs in x minutes.

b. Graph the data.

c. How far does the

cheetah run in 5

minutes?

y = 0.75x

Dis

tance

(m

i)

Time (h)

4

6

8

2 4 6

2

8

3.75 miles